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CSYSVX(3F)							    CSYSVX(3F)

NAME
     CSYSVX - use the diagonal pivoting factorization to compute the solution
     to a complex system of linear equations A * X = B,

SYNOPSIS
     SUBROUTINE CSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
			X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO )

	 CHARACTER	FACT, UPLO

	 INTEGER	INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS

	 REAL		RCOND

	 INTEGER	IPIV( * )

	 REAL		BERR( * ), FERR( * ), RWORK( * )

	 COMPLEX	A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
			LDX, * )

PURPOSE
     CSYSVX uses the diagonal pivoting factorization to compute the solution
     to a complex system of linear equations A * X = B, where A is an N-by-N
     symmetric matrix and X and B are N-by-NRHS matrices.

     Error bounds on the solution and a condition estimate are also provided.

DESCRIPTION
     The following steps are performed:

     1. If FACT = 'N', the diagonal pivoting method is used to factor A.
	The form of the factorization is
	   A = U * D * U**T,  if UPLO = 'U', or
	   A = L * D * L**T,  if UPLO = 'L',
	where U (or L) is a product of permutation and unit upper (lower)
	triangular matrices, and D is symmetric and block diagonal with
	1-by-1 and 2-by-2 diagonal blocks.

     2. The factored form of A is used to estimate the condition number
	of the matrix A.  If the reciprocal of the condition number is
	less than machine precision, steps 3 and 4 are skipped.

     3. The system of equations is solved for X using the factored form
	of A.

     4. Iterative refinement is applied to improve the computed solution
	matrix and calculate error bounds and backward error estimates
	for it.

									Page 1

CSYSVX(3F)							    CSYSVX(3F)

ARGUMENTS
     FACT    (input) CHARACTER*1
	     Specifies whether or not the factored form of A has been supplied
	     on entry.	= 'F':	On entry, AF and IPIV contain the factored
	     form of A.	 A, AF and IPIV will not be modified.  = 'N':  The
	     matrix A will be copied to AF and factored.

     UPLO    (input) CHARACTER*1
	     = 'U':  Upper triangle of A is stored;
	     = 'L':  Lower triangle of A is stored.

     N	     (input) INTEGER
	     The number of linear equations, i.e., the order of the matrix A.
	     N >= 0.

     NRHS    (input) INTEGER
	     The number of right hand sides, i.e., the number of columns of
	     the matrices B and X.  NRHS >= 0.

     A	     (input) COMPLEX array, dimension (LDA,N)
	     The symmetric matrix A.  If UPLO = 'U', the leading N-by-N upper
	     triangular part of A contains the upper triangular part of the
	     matrix A, and the strictly lower triangular part of A is not
	     referenced.  If UPLO = 'L', the leading N-by-N lower triangular
	     part of A contains the lower triangular part of the matrix A, and
	     the strictly upper triangular part of A is not referenced.

     LDA     (input) INTEGER
	     The leading dimension of the array A.  LDA >= max(1,N).

     AF	     (input or output) COMPLEX array, dimension (LDAF,N)
	     If FACT = 'F', then AF is an input argument and on entry contains
	     the block diagonal matrix D and the multipliers used to obtain
	     the factor U or L from the factorization A = U*D*U**T or A =
	     L*D*L**T as computed by CSYTRF.

	     If FACT = 'N', then AF is an output argument and on exit returns
	     the block diagonal matrix D and the multipliers used to obtain
	     the factor U or L from the factorization A = U*D*U**T or A =
	     L*D*L**T.

     LDAF    (input) INTEGER
	     The leading dimension of the array AF.  LDAF >= max(1,N).

     IPIV    (input or output) INTEGER array, dimension (N)
	     If FACT = 'F', then IPIV is an input argument and on entry
	     contains details of the interchanges and the block structure of
	     D, as determined by CSYTRF.  If IPIV(k) > 0, then rows and
	     columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1
	     diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then
	     rows and columns k-1 and -IPIV(k) were interchanged and D(k-
	     1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k)

									Page 2

CSYSVX(3F)							    CSYSVX(3F)

	     = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
	     interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

	     If FACT = 'N', then IPIV is an output argument and on exit
	     contains details of the interchanges and the block structure of
	     D, as determined by CSYTRF.

     B	     (input) COMPLEX array, dimension (LDB,NRHS)
	     The N-by-NRHS right hand side matrix B.

     LDB     (input) INTEGER
	     The leading dimension of the array B.  LDB >= max(1,N).

     X	     (output) COMPLEX array, dimension (LDX,NRHS)
	     If INFO = 0, the N-by-NRHS solution matrix X.

     LDX     (input) INTEGER
	     The leading dimension of the array X.  LDX >= max(1,N).

     RCOND   (output) REAL
	     The estimate of the reciprocal condition number of the matrix A.
	     If RCOND is less than the machine precision (in particular, if
	     RCOND = 0), the matrix is singular to working precision.  This
	     condition is indicated by a return code of INFO > 0, and the
	     solution and error bounds are not computed.

     FERR    (output) REAL array, dimension (NRHS)
	     The estimated forward error bound for each solution vector X(j)
	     (the j-th column of the solution matrix X).  If XTRUE is the true
	     solution corresponding to X(j), FERR(j) is an estimated upper
	     bound for the magnitude of the largest element in (X(j) - XTRUE)
	     divided by the magnitude of the largest element in X(j).  The
	     estimate is as reliable as the estimate for RCOND, and is almost
	     always a slight overestimate of the true error.

     BERR    (output) REAL array, dimension (NRHS)
	     The componentwise relative backward error of each solution vector
	     X(j) (i.e., the smallest relative change in any element of A or B
	     that makes X(j) an exact solution).

     WORK    (workspace/output) COMPLEX array, dimension (LWORK)
	     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The length of WORK.  LWORK >= 2*N, and for best performance LWORK
	     >= N*NB, where NB is the optimal blocksize for CSYTRF.

     RWORK   (workspace) REAL array, dimension (N)

     INFO    (output) INTEGER
	     = 0: successful exit
	     < 0: if INFO = -i, the i-th argument had an illegal value

									Page 3

CSYSVX(3F)							    CSYSVX(3F)

	     > 0: if INFO = i, and i is
	     <= N: D(i,i) is exactly zero.  The factorization has been
	     completed, but the block diagonal matrix D is exactly singular,
	     so the solution and error bounds could not be computed.  = N+1:
	     the block diagonal matrix D is nonsingular, but RCOND is less
	     than machine precision.  The factorization has been completed,
	     but the matrix is singular to working precision, so the solution
	     and error bounds have not been computed.

									Page 4

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